Prove2Me

A squared total times quadratic reciprocals is at least thirty-six

Proved
WorkbookSource.base_6511

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Prove that for positives xx, yy, and zz, the inequality (x+y+z)2(3x2+y2+z2+1x2+1y2+1z2)≥36(x+y+z)^2 \left(\frac{3}{x^2+y^2+z^2}+\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\right) \geq 36 holds.

Source: InternLM Lean-Workbook, record lean_workbook_6511 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_6511 (x y z : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (x + y + z) ^ 2 * (3 / (x ^ 2 + y ^ 2 + z ^ 2) + 1 / x ^ 2 + 1 / y ^ 2 + 1 / z ^ 2) ≥ 36  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_6511; Apache-2.0

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